Some Inequalities for Modified Bessel Functions - EMIS

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 253035, 10 pages doi:10.1155/2010/253035

Research Article Some Inequalities for Modified Bessel Functions Andrea Laforgia and Pierpaolo Natalini Department of Mathematics, Rome Tre University, L.go S. Leonardo Murialdo, 1- 00146 Rome, Italy Correspondence should be addressed to Pierpaolo Natalini, [email protected] Received 15 October 2009; Accepted 28 December 2009 Academic Editor: Ram N. Mohapatra Copyright q 2010 A. Laforgia and P. Natalini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We denote by Iν and Kν the Bessel functions of the first and third kind, respectively. Motivated by the relevance of the function wν t tIν−1 t/Iν t, t > 0, in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector 1984, we establish new inequalities for Iν t/Iν−1 t. The results are based on the recurrence relations for Iν and Iν−1 and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for Kν t/Kν−1 t.

1. Introduction Inequalities for modified Bessel functions Iν t and Kν t have been established by many authors. For example, Bordelon 1 and Ross 2 proved the bounds

ex−y

ν ν Iν x x x < < ey−x , y y Iν y

ν > 0, 0 < x < y.

1.1

The lower bound was also proved by Laforgia 3 for larger domain ν > −1/2. In 3 also the following bounds: y ν Iν x , < ex−y x Iν y

ν≥

1 , 0 < x < y, 2

1.2

y ν Kν x , < ey−x x Kν y

ν>

1 , 0 < x < y, 2

1.3

2

Journal of Inequalities and Applications y ν Kν x , > ey−x x Kν y

1 , 0 < x < y, 2

0 0, ν ≥ − , 2

Kν−1 tKν1 t > Kν2 t,

t > 0, ∀ν ∈ R

1.8 1.9

proved in 12, 13, respectively see also 14 for 1.9. Inequalities 1.8-1.9 have been used, recently, by Baricz in 15, to prove, in different way, the known inequalities

t t

Iν t 2 < t ν2 , Iν t

Kν t < − t2 ν 2 , Kν t

ν≥−

1 2

ν ∈ R.

The results are given by the following theorems.

1.10 1.11

Journal of Inequalities and Applications

3

Theorem 1.1. For real ν let Iν t be the modified Bessel function of the first kind and order ν. Then −ν

√

ν 2 t2 Iν t < , t Iν−1 t

ν ≥ 0.

1.12

In particular, for ν ≥ 1/2, the inequality Iν t/Iν−1 t < 1 holds also true. Theorem 1.2. For real ν let Kν t be the modified Bessel function of the third kind and order ν. Then Kν t ν < Kν−1 t

√

ν 2 t2 , t

∀ν ∈ R.

1.13

In particular, for ν > 1/2, the inequality Kν t/Kν−1 t > 1 holds also true.

2. The Proofs Proof of Theorem 1.1. The upper bound for the ratio Iν t/Iν−1 t follows from the inequality Iν t < Iν−1 t,

ν≥

1 2

2.1

proved by Soni for ν > 1/2 16, and extended by Näsell to ν 1/2 17. To prove the lower bound in 1.12, we substitute the function Iν1 t given by 1.6 in the Turán-type inequality 1.8. We get, for ν ≥ −1/2,

2ν Iν t < Iν2 t, Iν−1 t Iν−1 t − t

2.2

that is, 1−

I 2 t 2ν Iν t < 2ν . t Iν−1 t Iν−1 t

2.3

We denote Iν t/Iν−1 t by u and observe that for ν ≥ 1/2, by 2.1, u < 1. With this notation 2.3 can be written as u2

2ν u − 1 > 0, t

2.4

which gives, for ν ≥ 0, ν − t

ν2 1 < u, t2

2.5

4


that is,

−ν ν2 t2 Iν−1 t < tIν t

2.6

which is the desired result. Remark 2.1. For ν > 0, Jones 18 proved stronger result than 2.1 that the function Iν t decreases with respect to ν, when t > 0. Proof of Theorem 1.2. The proof is similar to the one used to prove Theorem 1.1. By Kν1 t > Kν t,

1 ν>− , 2

2.7

we get Kν t/Kν−1 t > 1, for ν > 1/2. We substitute the function Kν1 t given by 1.7 in 1.9. We get

2ν Kν t ≥ Kν2 t, Kν−1 t Kν−1 t t

∀ν ∈ R

2.8

or, equivalently 1

2ν Kν t − t Kν−1 t

Kν t Kν−1 t

2

≥ 0,

2.9

u

Kν t . Kν−1 t

2.10

ν 2 t2 , t

∀ν ∈ R

2.11

that is, u2 −

2ν u − 1 ≤ 0, t

Finally, we obtain Kν t ν < Kν−1 t

√

which is the desired result 1.13. Remark 2.2. By means the integral formula 11, page 181 Kν t

∞

e−t cosh z coshνzdz,

ν > −1,

2.12

0

follows immediately the inequality Kν−1 t > Kν t,

0 gν t we obtain, by simple calculations, the following one t1 − ν ν − 1/2 < 0 which is satisfied for all t > 1/2 − ν/1 − ν when ν > 1. We report here some numerical experiments, computed by using mathematica. Example 3.2. In the first case we assume ν 8. In Figure 1 we report the graphics of the functions Iν t/Iν−1 t solid line and the respective lower bounds fν t short dashed line and gν t long dashed line on the interval 100, 600. In Table 1 we report also the respective numerical values of the differences Iν t/Iν−1 t − fν t and Iν t/Iν−1 t − gν t in some points t. Remark 3.3. By some numerical experiments we can conjecture that the lower bound 3.1 holds true also when 1/2 ≤ ν < 1 and, in particular, for these values of ν we have fν t < gν t. See, for example, in Figure 2 the graphics of the functions Iν t/Iν−1 t solid line and the respective lower bounds fν t short dashed line and gν t long dashed line on the interval 100, 600 when ν 0.7.

6

Journal of Inequalities and Applications Table 1 t 600 0.00081226756 0.01195806306

Iν t/Iν−1 t − fν t Iν t/Iν−1 t − gν t

t 6000 0.00008312164 0.00124444278

t 60000 8.3312153 × 10−6 0.00012494429

Table 2 t 600 0.000854625

hν t − Kν t/Kν−1 t

t 6000 0.0000835453

t 60000 8.33545 × 10−6

0.98 0.96 0.94 0.92

200

300

400

500

600

400

500

600

0.88

Figure 1

0.999 0.998 0.997 0.996 0.995 0.994 200

300

Figure 2

Example 3.4. In this case we assume ν 1/3, then we report, in Figure 3, the graphics of the functions Iν t/Iν−1 t solid line and the respective lower bounds fν t short dashed line on the interval 100, 600. In Table 3 we report also the respective numerical values of the differences Iν t/Iν−1 t − fν t in some points t. Example 3.5. Also in this case we assume ν 8. In Figure 4 we report the graphics √ of the functions Kν t/Kν−1 t solid line and the respective upper bound hν t ν ν2 t2 /t short dashed line on the interval 100, 600. In Table 2, we report also the respective numerical values of the difference hν t − Kν t/Kν−1 t in some points t.


7 Table 3

t 600 0.00083345

Iν t/Iν−1 t − fν t

t 6000 0.0000833345

t 60000 8.33334 × 10−6

Table 4 t 600 0.000821238

hν t − Kν t/Kν−1 t

t 6000 0.0000832119

t 60000 8.33212 × 10−6

1.001

200

300

400

500

600

0.999 0.998 0.997

Figure 3

1.08 1.07 1.06 1.05 1.04 1.03 200

300

400

500

600

Figure 4

0.99 0.985 0.98 0.975 0.97 0.965 200

300

Figure 5

400

500

600

8

Journal of Inequalities and Applications Table 5 ν 0.3 0.109926 0.133826 0.195323

aν x, y bν x, y Iν x/Iν y

ν8 0.000528653 0.00272121 0.00282361

ν 30 1.26041 × 10−10 8.42928 × 10−10 8.45623 × 10−10

Table 6 ν 0.2 8.4878 7.42604 10.2446

cν x, y dν x, y Kν x/Kν y

ν 0.4 9.74992 7.53746 10.3615

ν8 — 367.483 384.34

ν 30 — 1.18634 × 109 1.19039 × 109

Example 3.6. In this last case we assume ν −4. In Figure 5 we report the graphics √ of the functions Kν t/Kν−1 t solid line and the respective upper bound hν t ν ν2 t2 /t short dashed line on the interval 100, 600. In Table 4 we report also the respective numerical values of the difference hν t − Kν t/Kν−1 t in some points t. Remark 3.7. We conclude this paper observing that, dividing by t inequalities 1.10-1.11 and integrating them from x to y 0 < x < y, we obtain the following new lower bounds for the ratios Iν x/Iν y and Kν x/Kν y: ⎛ ⎞ν ν ν ν 2 y 2 √ x ⎜ ⎟ √ν2 x2 − ν2 y2 Iν x < , √ ⎝ ⎠ e y Iν y ν ν 2 x2

y ν x

⎞ν √ √ 2 2 √ 2 2 K x 2 2 ⎜ν ν x ⎟ ν , ⎝ ⎠ e ν y − ν x < Kν y ν ν2 y2

1 ν≥− , 2

3.2

∀ν ∈ R.

3.3

⎛

For a survey on inequalities of the type 3.2 and 3.3 see 4. In the following Tables 5 and 6 we confront the lower bounds 1.1–3.2 and 1.4– 3.3, respectively, for different values of ν in the particular cases x 2 and y 4. Let

aν x, y

ν x ex−y , y

⎛ ⎞ν ν ν ν 2 y 2 √ x ⎜ ⎟ √ν2 x2 − ν2 y2 bν x, y , √ ⎝ ⎠ e y ν ν 2 x2


9

y ν y−x cν x, y e , x

dν x, y

y ν x

⎛

⎞ν √ √2 2 √2 2 2 x2 ν ν ⎜ ⎟ ⎝ ⎠ e ν y − ν x , ν ν2 y2 3.4

then we have Tables 5 and 6. By the values reported on Table 5 it seems that bν x, y is a lower bound much more stringent with respect to aν x, y for every ν > 0 moreover we recall that 3.2 holds true also for −1/2 < ν ≤ 0, while by the values reported on Table 6 it seems that cν x, y is a lower bound more stringent with respect to dν x, y for 0 < ν < 1/2 but we recall that 3.3 holds true also for ν ≤ 0 and ν ≥ 1/2.

Acknowledgment This work was sponsored by Ministero dell’Universitá e della Ricerca Scientifica Grant no. 2006090295.

References 1 D. J. Bordelon, “Solution to problem 72-15,” SIAM Review, vol. 15, pp. 666–668, 1973. 2 D. K. Ross, “Solution to problem 72-15,” SIAM Review, vol. 15, pp. 668–670, 1973. 3 A. Laforgia, “Bounds for modified Bessel functions,” Journal of Computational and Applied Mathematics, vol. 34, no. 3, pp. 263–267, 1991. ´ Baricz, “Bounds for modified Bessel functions of the first and second kind,” Proceedings of the 4 A. Edinburgh Mathematical Society. In press. ´ Baricz, “Tight bounds for the generalized Marcum Q-function,” Journal of Mathematical Analysis 5 A. and Applications, vol. 360, no. 1, pp. 265–277, 2009. ´ Baricz and Y. Sun, “New bounds for the generalized marcum Q-function,” IEEE Transactions on 6 A. Information Theory, vol. 55, no. 7, pp. 3091–3100, 2009. 7 J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Transactions on Information Theory, vol. 6, pp. 59–267, 1960. 8 J. I. Marcum and P. Swerling, “Studies of target detection by pulsed radar,” IEEE Transactions on Information Theory, vol. 6, pp. 227–228, 1960. 9 A. H. Nuttall, “Some integrals involving the QM function,” IEEE Transactions on Information Theory, vol. 21, pp. 95–96, 1975. 10 M. K. Simon and M. S. Alouini, Digital Communication over Fadding Channels: A Unified Approach to Performance Analysis, John Wiley & Sons, New York, NY, USA, 2000. 11 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, DC, USA, 1964. 12 L. Lorch, “Monotonicity of the zeros of a cross product of Bessel functions,” Methods and Applications of Analysis, vol. 1, no. 1, pp. 75–80, 1994. 13 M. E. H. Ismail and M. E. Muldoon, “ Monotonicity of the zeros of a cross-product of Bessel functions,” SIAM Journal on Mathematical Analysis, vol. 9, no. 4, pp. 759–767, 1978. 14 A. Laforgia and P. Natalini, “On some Turán-type inequalities,” Journal of Inequalities and Applications, vol. 2006, Article ID 29828, 6 pages, 2006. ´ Baricz, “On a product of modified Bessel functions,” Proceedings of the American Mathematical 15 A. Society, vol. 137, no. 1, pp. 189–193, 2009. 16 R. P. Soni, “On an inequality for modified Bessel functions,” Journal of Mathematical Physics, vol. 44, pp. 406–407, 1965.

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17 I. Näsell, “Inequalities for modified Bessel functions,” Mathematics of Computation, vol. 28, pp. 253– 256, 1974. 18 A. L. Jones, “An extension of an inequality involving modified Bessel functions,” Journal of Mathematical Physics, vol. 47, pp. 220–221, 1968. 19 H. C. Simpson and S. J. Spector, “Some monotonicity results for ratios of modified Bessel functions,” Quarterly of Applied Mathematics, vol. 42, no. 1, pp. 95–98, 1984. 20 H. C. Simpson and S. J. Spector, “On barrelling for a special material in finite elasticity,” Quarterly of Applied Mathematics, vol. 42, no. 1, pp. 99–111, 1984.